Question

A random sample of 390 married couples found that 298 had two or more personality preferences in common. In another random sample of 582 married couples, it was found that only 28 had no preferences in common. Let p₁ be the population proportion of all married couples who have two or more personality preferences in common. Let p₂ be the population proportion of all married couples who have no personality preferences in common.

(a) Find a 95% confidence interval for p₁ – p₂. (Round your answers to three decimal places.)

lower limit
upper limit

(b) Explain the meaning of the confidence interval in part (a) in the context of this problem. Does the confidence interval contain all positive, all negative, or both positive and negative numbers? What does this tell you (at the 95% confidence level) about the proportion of married couples with two or more personality preferences in common compared with the proportion of married couples sharing no personality preferences in common?

Because the interval contains only positive numbers, we can say that a higher proportion of married couples have two or more personality preferences in common.We can not make any conclusions using this confidence interval.    Because the interval contains both positive and negative numbers, we can not say that a higher proportion of married couples have two or more personality preferences in common.Because the interval contains only negative numbers, we can say that a higher proportion of married couples have no personality preferences in common.

Answer 1

A) = 298/390 = 0.764

   = 28/582 = 0.048

The pooled sample proportion (P) = ( * n1 + * n2)/(n1 + n2)

= (0.764 * 390 + 0.048 * 582)/(390 + 582)

= 0.335

SE = Sqrt(P(1 - P)(1/n1 + 1/n2))

= sqrt(0.335 * (1 - 0.335) * (1/390 + 1/582))

= 0.031

At 95% confidence interval the critical value is z^* = 1.96

The 95% confidence interval is

() +/- z^* * SE

= (0.764 - 0.048) +/- 1.96 * 0.031

= 0.716 +/- 0.061

= 0.655, 0.777

Because the interval cointains only positive numbers , we can say that a higher proportion of married couples have two or more personality preference in common.